Method and apparatus for representing complex vector data

ABSTRACT

A method and apparatus for representing a complex vector signal by determining representative coordinates for the complex vector signal, wherein at least some of the determined representative coordinates are associated with corresponding reference coordinates; associating each of at least a portion of the determined representative coordinates and corresponding reference coordinates with at least one of a plurality of temporal spans; and selectively processing each of a plurality of temporal spans.

Technical Field

[0001] The invention relates generally to signal analysis devices and,more specifically, to a method and apparatus for representing complexvector data.

BACKGROUND OF THE INVENTION

[0002] A common practice in the field of communication system analysisis to display a time-varying complex signal vector by plotting the realcomponent and imaginary component parametrically as a function of time,as Cartesian coordinates on a plane. Such a display is frequentlyreferred to as a polar display, independent of whether Cartesian orpolar axes are used. For example, using a representative complex signalS(t)=I(t)+j*Q(t), the values I(t) and Q(t) are both plottedparametrically against time.

[0003] Frequently it is of interest to simultaneously display areference waveform for comparison to a measured waveform. For example,the reference waveform might be a mathematically synthesized waveformthat is optimum according to some criteria. FIG. 1 graphically depicts areference waveform denoted as R(t) that is graphically overlaid with acorresponding measured waveform denoted as S(t), where the independentvariable t spans some range, say, t₁<t<t₂. In this example, S(t) isshown as a solid line and R(t) is shown as a dashed line. While thediagram of FIG. 1 allows direct comparison of the measured waveform S(t)with the reference waveform R(t), this diagram is only useful if thetime span, t₂−t₁, is relatively short. This is because as the time spanof the measured and corresponding reference waveforms increases, thediagram or display becomes too complex to be of practical use.Specifically, the signal or line density increases beyond a level atwhich a viewer may readily discern useful information. In such a case,the resulting display is useful only for gross qualitative assessment,since line density is so great that individual vector paths are hiddenwithin nearly coincident segments of the same waveform. Further, in thiscase, the use of reference waveforms is almost pointless, since thecorresponding increase in line density further compounds the problem ofvisually conveying useful information.

[0004] One method of utilizing a reference waveform within the contextof a long time duration is to subtract the reference waveform from themeasured waveform such that the magnitude of the resulting difference isdisplayed as a function of time. While such a waveform is useful infinding a point at which a maximum difference occurs, any correlationbetween the difference or errors and the position of the measuredwaveform in a complex plane is lost.

SUMMARY OF THE INVENTION

[0005] These and other deficiencies of the prior art are addressed bythe present invention of a method and apparatus for representing complexvector signals or data. The invention advantageously allows very longrecords (i.e., vector data including corresponding reference vector dataaccumulated over a relatively long time period) to be viewed in a mannerconveying to the viewer useful information.

[0006] A method for representing a complex vector signal according toone embodiment of the invention comprises determining representativecoordinates for the complex vector signal, wherein at least some of thedetermined representative coordinates are associated with correspondingreference coordinates; associating each of at least a portion of thedetermined representative coordinates and corresponding referencecoordinates with at least one of a plurality of temporal spans; andselectively processing each of a plurality of temporal spans.

BRIEF DESCRIPTION OF THE DRAWINGS

[0007] The teachings of the present invention can be readily understoodby considering the following detailed description in conjunction withthe accompanying drawings, in which:

[0008]FIG. 1 graphically depicts a prior art polar displayrepresentation of a complex vector signal and a corresponding referencesignal;

[0009]FIG. 2 depicts a high-level block diagram of a signal analysissystem;

[0010]FIG. 3 depicts a high-level block diagram of a controller suitablefor use in the signal analysis system of FIG. 2;

[0011]FIG. 4 depicts a graphical representation of a temporalsegmentation of a coordinate stream;

[0012]FIG. 5 graphically depicts a plurality of image frames;

[0013]FIG. 6 depicts a user interface adapted to presenting image framessuch as shown in FIG. 5;

[0014]FIG. 7 graphically depicts user interface functions suitable foruse in the user interface of FIG. 6; and

[0015]FIG. 8 depicts a flow diagram of a method according to anembodiment of the invention.

[0016] To facilitate understanding, identical reference numerals havebeen used, where possible, to designate identical elements that arecommon to the figures.

DETAILED DESCRIPTION OF THE INVENTION

[0017] The subject invention will be primarily described within thecontext of a measurement apparatus such as a digital storageoscilloscope (DSO) that receives at least one signal under test (SUT)that may be modeled as a complex vector that varies with time. It willbe appreciated by those skilled in the art that the SUT, illustrativelya communications signal having in-phase and quadrature components, maybe a real signal having limited bandwidth when represented in thefrequency domain, such that the signal takes on the interpretation of acomplex signal when added to its Hilbert transform.

[0018] The SUT is processed to derive a corresponding stream ofcoordinates, where each coordinate is a complex number consisting of areal part and an imaginary part (corresponding to the in-phase andquadrature components, respectively). The stream of coordinates isdivided into a plurality of segments, where each segment is associatedwith at least one coordinate. A plurality of logically sequential imageframes is formed by associating each of a sequence of image frames witha respective plurality of contiguous segments. The frames areselectively displayed upon a display device to present, thereby,respective temporal portions of the stream of coordinates representingthe signal under test. Optionally, coordinates derived from a referencesignal may be associated with the initially generated stream ofcoordinates such that the displayed frame(s) also depicts the referencesignal. It will be appreciated by those skilled in the art that theinvention may be advantageously employed in any signal measurement,analysis or display device in which time varying vector signalprocessing is employed.

[0019]FIG. 2 depicts a high-level block diagram of a signal analysisdevice. Specifically, the signal analysis device 10 of FIG. 2 receives asignal under test (SUT) from an input probe (not shown) and responsivelydisplays data generated using the SUT in the manner described below. TheSUT comprises, illustratively, a quadrature amplitude modulation (QAM),phase shift keyed (PSK) or other complex vector signal utilizing, forexample, a carrier signal having in-phase and quadrature phasecomponents.

[0020] The signal analysis device 10 of FIG. 2 comprises an analog todigital (A/D) converter 110, a clock source 130, an acquisition memory140, a controller 150, an input device 160 and a display device 170.

[0021] The A/D converter 110 receives and digitizes a signal under test(SUT) in response to a clock signal CLK produced by the clock source130. The clock signal CLK is preferably a clock signal adapted to causethe A/D converter 110 to operate at a maximum sampling rate, thoughother sampling rates may be selected. The clock signal 130 is optionallyresponsive to a clock control signal CC produced by the controller 150to change frequency and/or pulse width parameters associated with theclock signal CLK.

[0022] An output signals SUT′ produced by the A/D converter 110 isstored in the acquisition memory 140. The acquisition memory 140cooperates with the controller 150 to store data samples provided by theA/D converter 110 in a controlled manner such that samples from the A/Dconverter 110 may be provided to the controller 150 for furtherprocessing and/or analysis.

[0023] The controller 150 is used to manage the various operations ofthe signal analysis device 10. The controller 150 performs variousprocessing and analysis operations on the data samples stored within theacquisition memory 140. The controller 150 receives user commands via aninput device 160, illustratively a keypad or pointing device. Anembodiment of the controller 150 will be described in more detail belowwith respect to FIG. 3.

[0024] The signal analysis device 10 of FIG. 2 is depicted as receivingonly one signal under test (SUT). However, it will be appreciated bythose skilled in the art that many signals under test may be receivedand processed by the signal analysis device 10. Each signal under testis preferably processed using a respective A/D converter 110, which A/Dconverter may be clocked using the clock signal CLK provided by clocksource 130 or by another clock source. Additional memory may also beemployed to support the processing of additional signals under test.

[0025]FIG. 3 depicts a high-level block diagram of a controller suitablefor use in the signal analysis device 10 of FIG. 1. Specifically, thecontroller 150 of FIG. 3 comprises a processor 154 as well as memory 158for storing various control programs 158-P. The processor 154 cooperateswith conventional support circuitry 156 such as power supplies, clockcircuits, cache memory and the like as well as circuits that assist inexecuting the software routines stored in the memory 158. As such, it iscontemplated that some of the process steps discussed herein as softwareprocesses may be implemented within hardware, for example as circuitrythat cooperates with the processor 154 to perform various steps. Thecontroller 150 also contains input/output (I/O) circuitry 152 that formsan interface between the various functional elements communicating withthe controller 150. For example, in the embodiment of FIG. 2, thecontroller 150 optionally communicates with the clock source 130 (viaclock control signal CC). The controller 150 also communicates with theinput device 160 via a signal path IN, a display device 170 via a signalpath OUT and the acquisition memory 140 via a memory bus MB. Thecontroller 150 may also communicate with additional functional elements(not shown). It is noted that the memory 158 of the controller 150 maybe included within the acquisition memory 140, that the acquisitionmemory 140 may be included within the memory 158 of the controller 150or that some portion of each may be shared.

[0026] Although the controller 150 of FIG. 3 is depicted as ageneral-purpose computer that is programmed to perform various controlfunctions in accordance with the present invention, the invention can beimplemented in hardware as, for example, an application specificintegrated circuit (ASIC). As such, the process steps described hereinare intended to be broadly interpreted as being equivalently performedby software, hardware or a combination thereof.

[0027]FIG. 4 depicts a graphical representation of a temporalsegmentation of a coordinate stream. Specifically, FIG. 4 depicts acoordinate stream CS that has been temporally divided into a pluralityof segments (delineated by respective time periods t₀, t₁ . . .t_(n . . .) ) It is noted that the coordinate stream CS comprises thepolar or Cartesian coordinates associated with a complex vector signalprocessed by, for example, the signal analysis device 10 of FIG. 1. Eachof the time segments comprises at least one coordinate or set ofcoordinates. Preferably, each time segment comprises a plurality ofcoordinates, though the number of coordinates associated with each timesegment is limited to avoid visual clutter during the subsequent displayor presentation of the various coordinates, as will become clear withthe following discussion.

[0028]FIG. 4 also depicts a plurality of frames denoted as frames F4,F5, F6 and F7 (collectively denoted as frames F). Each of the frames Fis logically constructed using a plurality of time segments,illustratively five time segments. Each frame is also depicted asoverlapping a sequentially adjacent frame by one temporal segment. Itwill be appreciated by those skilled in the art that the frames F may beconstructed of more or fewer time segments, that the number of timesegments overlapping may be more or fewer than one, and that otherchanges may be made in the logical construction of frame size, durationand the like. For purposes of clarity, a difference quantity (A)comprises a temporal difference equal to the duration of one temporalsegment. While the coordinate stream CS is depicted as having aplurality of temporal segments of equal duration, though it will beappreciated by those skilled in the art that temporal segments ofdiffering durations may also be employed within the context of thepresent invention.

[0029] Each of the temporal segments is logically constructed to containa plurality of adjacent vector coordinates. Each of the frames F islogically constructed to contain a plurality of adjacent temporalsegments. Those vector coordinates contained by a frame are intended tobe presented on a display device as part of a corresponding image. Byselectively displaying each frame in a sequence of frames (e.g., frameF4, then frame F5, then frame F6, then frame F7 and so on), theresulting presentation on the display device comprises a temporallyconstrained portion of the vector coordinates forming the coordinatestream CS. In this manner, a viewer or user attempting to discernfeatures or characteristics associated with the underlying signal undertest may concentrate on specific temporal regions of the coordinatestream and thereby more clearly interpret the data necessary to theviewer or user's purpose.

[0030]FIG. 5 graphically depicts a plurality of image frames.Specifically, FIG. 5 denotes exemplary imagery associated with each ofthe frames F4-F7 discussed above with respect to FIG. 4. Each of theimages 510, 520, 530 and 540 depicted in FIG. 5 (i.e., those associatedwith, respectively, frames F4-F7) depicts a respective frame in atemporally aligned sequence of frames presenting on a display devicevector coordinates from the coordinate stream CS of FIG. 4. Aspreviously discussed, the frames temporally overlap such that adjacentframes include common coordinate points. However, as a sequence offrames is displayed in a temporally advancing manner, the temporallyolder coordinate points are not displayed in subsequent frames, whiletemporally newer coordinate points are initially displayed in subsequentframes.

[0031] In the sequence of frames depicted in FIG. 5, each frame hasassociated with it a vector plot comprising “actual” (e.g., measured)data S(t) and reference data R(t) used to form the displayed waveform.In addition, the displayed waveform has a coordinate comprising the mostrecent temporal coordinate (denoted as (a)) and a coordinate comprisingthe least recent or oldest displayed coordinate (denoted as (b)). It canbe seen that the newest coordinate (a) “advances” over time from frameF4 through frame F7, while the oldest coordinate (b) “retreats” overtime from frames F4 through frame F7. Thus, the sequence of framesdepicted in FIG. 5 are time-banded by a temporal difference representedby the most recent coordinate (a) and the oldest coordinate (b).

[0032] Depending upon the complexity of the signal under test to beviewed, the amount of information displayed within one frame may beincreased or decreased. Such increase or decrease may be effected by,for example, increasing or decreasing the number of coordinates withineach time segment, or increasing or decreasing the number of timesegments within each frame. Additionally, the “persistence”characteristic of the waveform display may be increased or decreased byadapting the number of temporal segments overlapping between adjacentframes.

[0033] The imagery discussed above with respect to FIGS. 45 may becaptured and displayed on a measurement device such as a digital storageoscilloscope (DSO) or computer. Advantageously, the imagery 510 providesrapid assessment of complex signal density within regions of interest,such as specific constellation points depicted. It will be appreciatedthat more or fewer than eight constellation points may be depicted,depending upon the type of modulation employed to produce the signalunder test processed according to the teachings of the presentinvention. The inventors contemplate that the subject invention may beadapted to many different modulation techniques and many differentparameter combinations. Moreover, a user input may be received thatselects one or more displayed signal coordinates such that numeric (orother) parameters of the selected coordinate(s) are also displayed.

[0034]FIG. 6 depicts a user interface adapted to presenting image framessuch as shown in FIG. 5. Specifically, the user interface 600 of FIG. 6comprises a display region 620 having presented therein a first waveformS(t) generated using polar coordinates provided by, for example, thecoordinate stream CS discussed above with respect to FIGS. 4 and 5.Additionally, an optional reference waveform R(t) is presented in thedisplay region 620. The optional reference waveform R(t) is generatedby, for example, the processor 150 of the measuring system 10 of FIG. 1.The reference waveform R(t) comprises a waveform representing an idealor desired waveform to which the actual or measured waveform S(t) shouldconform. Deviations between the reference waveform R(t) and actualwaveform S(t) comprise errors or deviations that are useful inunderstanding the operation of a device or function producing theinitial signal under test.

[0035] The display region 600 displays imagery associated with a singleframe within a sequence of frames, such as discussed above with respectto FIGS. 4 and 5. The user interface 600 facilitates manipulation ofstored frames such that a user can perform various measurement andanalysis functions. Specifically, the user interface 600 of FIG. 6 alsocomprises a slider bar 610 including a temporal decrement icon 612, atemporal increment icon 614, a temporal location item 618 and a temporalrange region 616. The temporal advance icon 614, when selected by aninput device or other control input, results in the presentation withinthe display region 620 of the next frame within the sequence of frames.Similarly, selection of the temporal decrement icon 612 results in thepresentation of the previous frame within the sequence of frames. Therelative position of the temporal position icon 618 within the region616 indicates the relative position of the presently presented frame tothe other frames forming the sequence of frames (in this case theinitial frame is leftmost and the final frame is rightmost).

[0036]FIG. 7 graphically depicts user interface functions suitable foruse in the user interface of FIG. 6. Specifically, FIG. 7 depictsadditional icons or representations of functions that may be utilizedwithin the context of a user interface, such as discussed above withrespect to FIG. 6. Specifically, FIG. 7 depicts a rewind icon 710, a runbackward icon 720, a stop icon 730, a run-forward icon 740, and ago-to-end icon 750. The rewind icon is associated with a function that,upon selection, results in the first frame within a sequence of framesbeing displayed. The run-backward icon is associated with a functionthat, when selected, results in the sequential display in a reversetemporal order of the frames within the sequence of frames, beginningwith the presently displayed frame. The stop icon causes the continualdisplay the presently displayed frame. The run-forward icon 740 isassociated with a function that, when selected, causes the sequence offrames to be sequentially displayed beginning with the presentlydisplayed frame. The go-to-end icon 750 is associated with a functionthat, when selected, causes the final frame within the sequence offrames to be displayed.

[0037]FIG. 7 also depicts several icons associated with functionsadapted to modify the imagery to be displayed for each frame.Specifically, selectable icons associated with an increase sub-sequencefunction 760, a reduce sub-sequence function 770 and a selectsubsequence function 780 are provided. The increase subsequence function760 is used to increase the number of temporal segments forming eachframe (add more data to each frame). The reduce sub-sequence function770 is used to reduce the number of temporal segments forming each frame(reduce the data in each frame). The select sub-sequence function 770 isused to enter an interactive menu whereby a specific number of temporalsegments forming each frame may be selected. While not shown, additionalfunctions may also be provided. For example, additional icons associatedwith the functions of increasing, reducing and/or selecting the numberof vector coordinates associated with each temporal segment may also beprovided.

[0038]FIG. 8 depicts a flow diagram of a method according to anembodiment of the invention. Specifically, FIG. 8 depicts a flow diagramof a method for controllably displaying complex vector signal data and,optionally, corresponding reference signal data in a time-constrainedmanner.

[0039] The method 800 of FIG. 8 is entered at step 810 where coordinatesof the received complex vector signal are determined. This determinationmay be made by the controller 150 using samples stored in theacquisition memory 140 in the measurement system 10 of FIG. 1. Thecoordinates determined at step 810 represent an actual waveform S(t),such as noted in box 815. Determination of the coordinates optionallydepends on a supplied set of signal parameters, such as carrierfrequency, symbol rate, filter shape and bandwidth, or modulation type,such as noted in box 818. It is also noted that the coordinates maycomprise Cartesian coordinates, polar coordinates or other coordinatesuseful in performing the measurement task desired.

[0040] At step 820, at least a portion of the determined coordinates ofstep 810 are associated with corresponding reference coordinates. Thereference coordinates are those coordinates associated with an ideal ordesired waveform R(t), as denoted by box 825. The reference coordinatesoptionally depend on the supplied set of signal parameters of box 818.

[0041] At step 830, the sequence of determined coordinates from steps810 and, optionally, step 820, are temporally divided into a pluralityof time segments. Each time segment may have a fixed or variableduration At, as noted in box 835 and previously discussed.

[0042] At step 840, each time segment is associated with at least oneframe to form a sequence of temporally overlapping frames, as previouslydiscussed.

[0043] At step 850, the vector coordinates and, optionally, referencecoordinates, of each frame within a subsequence of frames are presentedon a display device, such as discussed above with respect to FIGS. 4-6.It is noted that one frame at a time may be displayed such that usermanipulation may cause a frame increment or frame decrement within thesequence of frames. The frames may also be displayed at a predefinedrate, such as a slow motion, fast motion or actual motion display rateas compared with the underlying signal under test.

[0044] At optional step 860, the size of frame sub-sequence or frameswithin the sub-sequence is adapted in response to control input. Thatis, as previously discussed with respect to FIG. 7, and noted in box865, the number of temporal segments within each frame may beincremented, decremented, selected or otherwise determined.

[0045] At optional step 870, the size of the time segment or number ofvector coordinates contained within a time segment is adapted inresponse to control input. That is, as previously discussed with respectto FIG. 7, the size of a time segment may be increased, decreased,selected or otherwise determined.

[0046] The above-described method may be used to process real-time dataor process previously stored data. Moreover, the processing stepsdiscussed herein may be performed in a different order, continuouslyand/or in parallel. For example, in one embodiment of the invention,memory within the signal measuring device 10 is used to store aplurality of digitized samples associated with a specific triggeredevent. The stored digitized samples are then processed to produce aplurality of complex vector data points, such as depicted above in thecoordinate stream CS of FIG. 4. In this embodiment, complex vectorsassociated with a finite amount of time are stored and subsequentlyprocessed and/or viewed to enable analysis of the underlying signalunder test. In a near real-time embodiment, memory within the signalmeasuring device 10 is increasingly utilized since data is beingacquired faster than the data is being presented and subsequentlydiscarded. In this embodiment, the memory within the signal measuringdevice 10 operates as a buffer that eventually constrains the imageryviewed by a user.

[0047] In another embodiment of the invention, the signals S(t) and R(t)from boxes 815 and 825 may be two simultaneously-acquired signals, suchas the input and output of a device, or the outputs of a device undertest and a reference device. One or both of such signals may also bedelayed in time to align relevant signal portions.

[0048] The reference signal may be used to delineate a referencecoordinate(s) or coordinate(s) range(s). The reference signal may alsobe derived from ideal or expected reference coordinate(s) range(s)stored in a memory. A combination of stored and received (i.e., derived)reference coordinate(s) or coordinate range(s) may also be used.

[0049] The above-described invention operates to display vectorcoordinates associated with a signal under test in a time-constrainedmanner such that a display remains visually uncluttered and, therefore,a user or a viewer of that display may derive useful information about asignal under test. Optionally, reference vector coordinates may also bedisplayed contemporaneously with the underlying actual vectorcoordinates such that deviations between actual and referencecoordinates may be readily appreciated by the viewer or user. In variousembodiments, the amount of data to be displayed at one time may bemodified by adapting the number of coordinates contained within a timesegment, the number of time segments contained within a frame, theamount by which adjacent frames overlap and/or other techniquesdiscussed above and appreciated by those skilled in the art informed bythe teachings of the present invention.

[0050] Although various embodiments which incorporate the teachings ofthe present invention have been shown and described in detail herein,those skilled in the art can readily devise many other variedembodiments that still incorporate these teachings.

What is claimed is:
 1. A method for representing a complex vectorsignal, comprising: determining representative coordinates for saidcomplex vector signal, wherein at least some of said determinedrepresentative coordinates are associated with corresponding referencecoordinates; associating each of at least a portion of said determinedrepresentative coordinates and corresponding reference coordinates withat least one of a plurality of temporal spans; and selectivelyprocessing each of a plurality of temporal spans.
 2. The method of claim1, wherein each of said temporal spans has an initial temporal valuethat is temporally separated from an initial temporal value of anadjacent temporal span by a predefined time increment.
 3. The method ofclaim 1, wherein said processing of a temporal span comprises displayingsaid determined representative coordinates and corresponding referencecoordinates associated with said temporal span.
 4. The method of claim3, wherein: said complex signal is logically divided into a plurality ofoverlapping temporal spans, each of said temporal spans adapted torepresent a respective image frame in a corresponding sequence of imageframes; and said selectively processing comprises displaying saidsequence of image frames according to a control signal.
 5. The method ofclaim 4, further comprising displaying indicium of the relative positionwithin said sequence of overlapping temporal spans of a temporal spanrepresenting a presently displayed image frame.
 6. The method of claim5, wherein said control signal is derived from an input signalindicative of a manipulation of said displayed indicium of relativeposition.
 7. The method of claim 4, wherein said control signalindicates one of a desired STOP, FORWARD and REVERSE operation.
 8. Themethod of claim 4, wherein said control signal indicates one of adesired GOTOEND and GOTOBEGINNING operation.
 9. The method of claim 1,wherein said determined representative coordinates and correspondingreference coordinates are displayed as a complex vector amplitude as afunction of time.
 10. The method of claim 1, wherein said referencecoordinates are determined using a second input signal.
 11. The methodof claim 1, wherein said reference coordinates represent at least one ofan expected coordinate and an expected range of coordinates.
 12. Themethod of claim 1, wherein said determined representative coordinatesand corresponding reference coordinates are displayed as amplitudes ofin-phase and quadrature components of said complex vector as a functionof time.
 13. The method of claim 12, wherein said amplitudes of in-phaseand quadrature components of said complex vector are plotted as aparametric function of time.
 14. The method of claim 1, wherein saiddetermined representative coordinates and corresponding referencecoordinates are displayed using one of a Cartesian coordinate system anda polar coordinate system.
 15. The method of claim 1, wherein eachtemporal span within a sequence of temporal spans is processed in asequential manner.
 16. The method of claim 15, wherein said sequentialprocessing of said temporal spans is adapted in response to a controlsignal.
 17. The method of claim 4, further comprising: adapting theextent that said temporal spans overlap each other in response to areceived control signal.
 18. The method of claim 1, further comprising:adapting the number of determined representative coordinates andcorresponding reference coordinates associated with each temporal spanin response to a control signal.
 19. The method of claim 2, furthercomprising: adjusting said predefined time increment in response to acontrol signal.
 20. The method of claim 3, further comprising:displaying said determined representative coordinates and correspondingreference coordinates associated with said temporal span within an imageregion of a user interface; said user interface further comprisinggraphical icons associated with frame display manipulation functions.21. The method of claim 20, wherein said frame display manipulationfunctions comprise at least one of a stop function, a forward functionand a reverse function.
 22. The method of claim 21, wherein saidfunction further includes at least one of a go-to-end function and ago-to-beginning function.
 23. A method for representing a complex vectorsignal, comprising: determining polar coordinates for said complexvector signal, said polar coordinates defining respective locations on apolar plane, wherein at least some of said determined polar coordinatesare associated with corresponding reference coordinates; associatingeach of at least a portion of said determined polar coordinates andcorresponding reference coordinates with at least one of a plurality oftemporal spans; and selectively processing each of a plurality oftemporal spans.
 24. The method of claim 23, wherein said processing of atemporal span comprises displaying said determined polar coordinates andcorresponding reference coordinates associated with said temporal span.25. Apparatus, comprising: means for determining representativecoordinates for said complex vector signal, wherein at least some ofsaid determined representative coordinates are associated withcorresponding reference coordinates; means for associating each of atleast a portion of said determined representative coordinates andcorresponding reference coordinates with at least one of a plurality oftemporal spans; and means for selectively processing each of a pluralityof temporal spans.
 26. A computer readable medium for storinginstructions which, when executed by a processor, perform the steps of:determining representative coordinates for a complex vector signal,wherein at least some of said determined representative coordinates areassociated with corresponding reference coordinates; associating each ofat least a portion of said determined representative coordinates andcorresponding reference coordinates with at least one of a plurality oftemporal spans; and selectively processing each of a plurality oftemporal spans.